Answer
$\begin{align}
& \left[ \sqrt{2}\left( \cos 45{}^\circ +i\sin 45{}^\circ \right) \right]\left[ 2\left( \cos 300{}^\circ +\sin 300{}^\circ \right) \right]\left[ 2\left( \cos 150{}^\circ +i\sin 150{}^\circ \right) \right]; \\
& 4\sqrt{2}\left( \cos 135{}^\circ +i\sin 135{}^\circ \right);\approx -4.0+4.0i \\
\end{align}$
Work Step by Step
Consider the expression
$z=\left( 1+i \right)\left( 1-i\sqrt{3} \right)\left( -\sqrt{3}+i \right)$
First, convert it into polar form:
$\begin{align}
& z=\left( 1+i \right)\left( 1-i\sqrt{3} \right)\left( -\sqrt{3}+i \right) \\
& =\left[ \sqrt{2}\left( \cos 45{}^\circ +i\sin 45{}^\circ \right) \right]\left[ 2\left( \cos 300{}^\circ +\sin 300{}^\circ \right) \right]\left[ 2\left( \cos 150{}^\circ +i\sin 150{}^\circ \right) \right]
\end{align}$
Apply the multiplication rule of complex numbers:
$\begin{align}
& z=\left( \sqrt{2}\times 2\times 2 \right)\left( \cos \left( 45{}^\circ +300{}^\circ +150{}^\circ \right)+i\sin \left( 45{}^\circ +300{}^\circ +150{}^\circ \right) \right) \\
& =4\sqrt{2}\left( \cos 495{}^\circ +i\sin 495{}^\circ \right) \\
& =4\sqrt{2}\left( \cos 135{}^\circ +i\sin 135{}^\circ \right)
\end{align}$
The above expression is the polar form of the provided expression.
Now, convert it into rectangular form:
$z=4\sqrt{2}\left( \cos 135{}^\circ +i\sin 135{}^\circ \right)$
Substitute the value of $\cos 135{}^\circ $ and $\sin 135{}^\circ $ in the above expression:
$\begin{align}
& z=4\sqrt{2}\left( -0.7071+i0.7071 \right) \\
& =-3.9999+i3.9999
\end{align}$
The above expression is in rectangular form of the provided expression is:
Polar form of the provided expression is $4\sqrt{2}\left( \cos 135{}^\circ +i\sin 135{}^\circ \right).$
Rectangular form of the provided expression is $\approx -4.0+4.0i.$
